Answer
$\dfrac{1}{30}$
Work Step by Step
Consider $I=\iiint_T xz dV$
$I= \int_{0}^1 \int_{0}^{x} \int_{0}^{x-y} xz dz dy dx= \int_{0}^1 \int_{0}^{x-y} [\dfrac{xz^2}{2}]_{0}^{x-y} dy dx$
or, $=\int_{0}^1 \int_{0}^{x}x[\dfrac{(x-y)^2}{2}] dy dx$
or, $= \int_{0}^{1}\int_0^x \dfrac{1}{2}[x(x^2-y^2+2xy)] dy dx$
or, $=\dfrac{1}{2} \int^{0}_{1} [\dfrac{x^3}{3}-\dfrac{xy^3}{3}+\dfrac{2x^2y^2}{2}]]_0^x dx$
or, $=(\dfrac{1}{2})[\dfrac{x^6}{15}]_0^1$
or, $\iiint_T xz dV=\dfrac{1}{30}$
